The way this wiki is set up, students must always log in before editing is allowed. Unlike Wikipedia, nobody here can edit anonymously. If a student "vandalizes" a page, you can block them and erase their stuff. If a student makes an honest mistake and damages a page while editing, you can easily recover the earlier undamaged version.
Everybody, including students, can examine user contributions. The best place to do so is either the "history" tab at the top of each page or "Special:Recentchanges" in the menu at left. Once you find the contribution you wish to see, click on "diff" which shows the changes which were made from the previous version.
For an example, follow this link. The yellow background on the left shows deleted material, while the green on the right is text added. For users in an academic course, a quick look at the history of a page quickly reveals who has been contributing. You can ask students to leave a comment when they make a change, and mark if it was a minor one.
To see how Reggie H. altered Steve W.'s prose see this example